(a) To prove that there exists a neighborhood W of the origin such that f(x) ≤ y for all x ∈ W, given that f is bounded above on a neighborhood V of the origin, we can use the linearity of f.
Since f is a linear function, it satisfies the following properties:
f(0) = 0
f(rx) = rf(x) for any scalar r and vector x
f(x + y) = f(x) + f(y) for any vectors x and y
Given that f is bounded above on V, there exists a positive number M such that f(x) ≤ M for all x ∈ V. Now, let's consider the neighborhood W defined as follows:
W = {x ∈ X | ||x|| < M}
We claim that for any x ∈ W, f(x) ≤ y.
Let x ∈ W. Since x is in the neighborhood W, we have ||x|| < M. By linearity, we can express x as x = rx' for some scalar r and vector x' with ||x'|| = 1.
Now, consider f(x):
f(x) = f(rx') = rf(x')
Since ||x'|| = 1, we have ||rx'|| = |r| ||x'|| = |r|.
Therefore, ||rx'|| < M implies |r| < M.
Using the fact that f is bounded above on V, we have f(x') ≤ M.
Combining these results, we get:
|f(x)| = |rf(x')| = |r| |f(x')| ≤ M
Since this inequality holds for any x ∈ W and |r| < M, we have shown that f(x) ≤ y for all x ∈ W, where W is a neighborhood of the origin.
(b) To prove that f is continuous, we can show that f is bounded above on any compact set in X. Let K be a compact set in X.
Since K is compact, it is also closed and bounded. By the linearity of f, we have:
f(K) = {f(x) | x ∈ K}
Since K is bounded, there exists a positive number M such that ||x|| ≤ M for all x ∈ K. By the linearity of f, we have:
f(K) = {f(x) | x ∈ K} ⊆ {f(x) | ||x|| ≤ M}
Thus, f(K) is bounded above by M.
By the previous result in part (a), if f is bounded above on a neighborhood of the origin, then it is bounded above on any neighborhood of the origin. Therefore, f is bounded above on the neighborhood V of the origin.
Since K is compact, it can be covered by finitely many neighborhoods of the origin, say V1, V2, ..., Vk. Thus, f is bounded above on each Vi, i = 1, 2, ..., k.
Now, consider the open cover {V1, V2, ..., Vk} of K. By compactness, there exists a finite subcover {V1, V2, ..., Vm}. Therefore, f is bounded above on K.
Since f is bounded above on any compact set K, it follows that f is continuous.
(c) The previous part (b) already proves that if f is bounded above on any compact set, it is continuous. Therefore, if f is bounded above on a set 2 with int(2) ≠ Ø (i.e., the interior of 2 is not empty), then f is continuous.
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why do i have depression
Answer:
are u ok tho
Step-by-step explanation:
What is the surface area of the prism below?
A
216 ft2
B
312 ft2
C
432 ft2
D
10,800 ft2
Complete the table of values y = x2 + 4x - 6
Answer:
x=-2
Step-by-step explanation:
number 3 please help me
Answer:
D. {1}Explanation:
given f(x) = 3x - 1
for f(x) = 2
3x - 1 = 2
3x = 2 + 1
3x = 3
x = 3/3
x = 1
the cost of a catering prom is 250 plus $15 for each person attending. what is the cost of 150 people?
Answer:
$2500
Step-by-step explanation:
First, multiply the $15 for the 150 people attending, to get 2250. Next, add the last $250 to get a total cost of $2500
Hopefully this helps- let me know if you have any questions!
Answer:
i think it is 2500
Step-by-step explanation:
take 15 times the amount of people. then add the 250 fee
In a correlated t test, if the independent variable has no effect, the sample diff ores are a random sample from a population where the mean difference score (μ d) equals______
a. 0 b. 1 c. N d. cannot be determined
The correct answer i.e. mean difference is (a) 0.
What is the mean difference?
The mean difference is a statistical measure that represents the average difference between pairs of values in a dataset. It is calculated by taking the sum of all the differences and dividing it by the total number of pairs.
To calculate the mean difference, follow these steps:
Identify the pairs of values in your dataset for which you want to calculate the difference.
Calculate the difference between each pair of values.
Sum up all the differences.
Divide the sum by the total number of pairs.
In a correlated t-test, the null hypothesis assumes that the mean difference between paired observations is zero, indicating no effect of the independent variable. Therefore, if the independent variable has no effect, the sample difference scores are expected to be a random sample from a population where the mean difference score, denoted as μd, equals 0.
Hence, the correct answer is (a) 0.
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find the general solution of the indicated differential equation. If possible, find an explicit solution. 1. y =xy 2. xy =2y 3. y =e
x−y 4. y =(1+y 2)e x5. y =xy+y 6. y =ye x −2e x +y−2 7. y =x/(y+2) 8. y =xy/(x−1) 9. x 2 y =ylny−y 10. xy −y=2x 2y 11. y 3y =x+2y 12. y =(2xy+2x)/(x 2−1)
The equations you mentioned:
1. y = xy:
To solve this differential equation, we can separate the variables and integrate both sides.
dy/y = x dx
Integrating both sides gives:
ln|y| = (1/2)x² + C
Exponentiating both sides gives the general solution:
|y| = [tex]e^{((1/2)x^2 + C)[/tex]
Taking the positive and negative values of y, we get two branches of solutions:
y = [tex]Ae^{(1/2)x^2[/tex] and y = [tex]-Ae^{(1/2)x^2[/tex], where A is an arbitrary constant.
2. xy = 2y:
Rearranging the equation, we get:
xy - 2y = 0
Factoring out y, we have:
y(x - 2) = 0
This equation has two solutions:
y = 0 and x - 2 = 0, which leads to x = 2.
3. y = [tex]e^x[/tex] - y:
Rearranging the equation, we get:
y + y = [tex]e^x[/tex]
Combining like terms, we have:
2y = [tex]e^x[/tex]
Dividing both sides by 2, we get:
y = (1/2)[tex]e^x[/tex]
4. y = (1 + y²)[tex]e^x[/tex]:
Rearranging the equation, we get:
y - y² = [tex]e^x[/tex]
Factoring out y, we have:
y(1 - y) = [tex]e^x[/tex]
This equation has two solutions:
y = 0 and 1 - y = [tex]e^x[/tex], which leads to y = 1 - [tex]e^x[/tex].
5. y = xy + y:
Rearranging the equation, we get:
y - xy - y = 0
Combining like terms, we have:
-xy = 0
This equation has two solutions:
x = 0 and y = 0.
6. y = y[tex]e^x[/tex] - 2[tex]e^x[/tex] + y - 2:
Rearranging the equation, we get:
y[tex]e^x[/tex] - y = 2[tex]e^x[/tex]- 2
Factoring out y, we have:
y([tex]e^x[/tex] - 1) = 2([tex]e^x[/tex] - 1)
Dividing both sides by ([tex]e^x[/tex] - 1), we get:
y = 2
7. y = x/(y + 2):
Rearranging the equation, we get:
y(y + 2) = x
Expanding the equation, we have:
y² + 2y - x = 0
This equation doesn't have a general solution in terms of elementary functions. It can be solved numerically or using approximation methods.
8. y = xy/(x - 1):
Rearranging the equation, we get:
(x - 1)y = xy
Dividing both sides by y and rearranging, we have:
x - 1 = x/y
Solving for y, we get:
y = x/(x - 1)
9. x²y = ylny - y:
This is a nonlinear differential equation that doesn't have a general solution in terms of elementary functions. It can be solved numerically or using approximation methods.
10. xy - y = 2x²y:
Rearranging the equation, we get:
xy - 2x²y - y = 0
Factoring out y, we have:
y(x - 2x² - 1) = 0
This equation has two solutions:
y = 0 and x - 2x² - 1 = 0, which leads to x = (1 ± √3)/2.
11. y - 3y² = x + 2y:
Rearranging the equation, we get:
-3y² + y + 2y - x = 0
Combining like terms, we have:
-3y² + 3y - x = 0
This equation doesn't have a general solution in terms of elementary functions. It can be solved numerically or using approximation methods.
12. y = (2xy + 2x)/(x² - 1):
Rearranging the equation, we get:
y(x² - 1) = 2xy + 2x
Expanding the equation, we have:
x²y - y = 2xy + 2x
Combining like terms, we get:
x²y - 2xy - y - 2x = 0
This equation doesn't have a general solution in terms of elementary functions. It can be solved numerically or using approximation methods.
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In the domain of all penguins, let D(x) be the predicate "x is dangerous." Translate the following quantified statement into simple, everyday English.
(∃x)¬D(x)
Step-by-Step solution please.
The quantified statement (∃x)¬D(x) can be translated into simple, everyday English as "There exists a penguin that is not dangerous."
The quantified statement (∃x)¬D(x) can be further explained in the context of penguins. Let's break it down:
The symbol (∃x) denotes the existence of an object or entity that satisfies a certain condition. In this case, it refers to a penguin that meets the condition specified afterward.
The predicate ¬D(x) can be understood as the negation of the predicate D(x), where D(x) represents the statement "x is dangerous." The negation symbol (¬) in front of D(x) indicates the opposite or negation of the statement.
Combining these elements, the quantified statement (∃x)¬D(x) asserts that there is at least one penguin for which the predicate "is not dangerous" holds true. In everyday English, this statement can be translated as "There exists a penguin that is not dangerous."
Essentially, it implies that within the domain of all penguins being considered, at least one penguin can be found that is not considered dangerous. This quantified statement allows for the possibility that not all penguins are dangerous and acknowledges the existence of non-threatening penguins.
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match each hypotenuse with the leg that will create a right
triangle
Answer:
Where is the question so that I can help you with it
10.
Consider the two properties that you would use to solve an equation like 3x + 5 = 26. Which of the following is true?
A. The standard method for solving an equation like 3x + 5 = 26 is to use the Multiplication Property of Equality and then the Division Property of Equality.
B. The standard method for solving an equation like 3x + 5 = 26 is to use the Division Property of Equality and then the Subtraction Property of Equality.
C. The standard method for solving an equation like 3x + 5 = 26 is to use the Subtraction Property of Equality and then the Division Property of Equality.
D. The standard method for solving an equation like 3x + 5 = 26 is to use the Subtraction Property of Equality and then the Addition Property of Equality.
Answer:
The standard method for solving an equation like 3x + 5 = 26 is to use the Subtraction Property of Equality and then the Division Property of Equality.
Step-by-step explanation:
To solve, you subtract both sides by 5 first. Then you divide both sides by 3 to isolate the x.
What value does the 2 represent in the number 0.826
Answer:
.02
Step-by-step explanation:
Soojeong estimates that it is going to take $180,000 to send her new born baby to college. She would like to start an annuity so that she can have $180,000 after 18 years. If the account earns 5.5% interest compounded monthly, how much must she put in each month?
Answer:
Monthly deposit= $489.59
Step-by-step explanation:
Giving the following information:
Future Value (FV)= $180,000
Number of periods (n)= 18*12= 216
Interest rate (i)= 0.055 / 12= 0.004583
To calculate the monthly deposit, we need to use the following formula:
FV= {A*[(1+i)^n-1]}/i
A= monthly deposit
Isolating A:
A= (FV*i)/{[(1+i)^n]-1}
A= (180,000*0.004583) / {[(1.004583)^216] - 1}
A= $489.59
A class interval refers to:
a) the number of categories within a group of data
b) a division used for grouping a set of observations
c) the mean of the set of data
d) the range of ages among a group of students
A class interval refers to option b) a division used for grouping a set of observations.
The correct answer is (b) a division used for grouping a set of observations. In statistics, when dealing with a large set of data, it is often helpful to group the data into intervals or classes to better understand the distribution. A class interval represents a range of values that are grouped together. It is defined by specifying the lower and upper boundaries of each interval.
For example, if we are analyzing the heights of individuals, we may create class intervals such as 150-160 cm, 160-170 cm, and so on. The purpose of using class intervals is to simplify the data and provide a clearer picture of the distribution. It allows us to summarize the data and identify patterns or trends within specific ranges. Therefore, option (b) is the correct description of a class interval.
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The price of calculator is increased from R150 to R174.What is the percentage increased
Answer:
16%
Step-by-step explanation:
(174-150)/150= 0.16
0.16*100 = 16%
Find the next two terms in
the sequence 11, 7, 3,-1,
Answer:
The next term will be -5.
Step-by-step explanation:
This is actually an AP with a = 11 and common difference d = -4.
Therefore next term will be -1 - 4 = -5
Answer:
Step-by-step explanation:
The numbers in this sequence come by subtracting 4 so
11, 7, 3, - 1, - 5, - 9, - 13
help plz ? Mark brainliest?
Answer:
-1(3x^2) +4x-7
Step-by-step explanation:
grouping
Matt tried to evaluate 49 x 24 using partial products. His work is shown below
Answer:
step 5
Step-by-step explanation:
Please help I really need it, I don’t know what to do
Answer:
i think it is b
Step-by-step explanation:
Which of the following is equal to the expression listed below?
18 + 12
OA. 6(3+2)
OB. (6 x 3)(6 x 2)
OC. 6+ (3 x 2)
OD. (6 + 3)(6 + 2)
Answer:
OA
Step-by-step explanation:
18+12=30
6×3=18
6×2=12
18+12=30
hope this helps
Answer:
Which of the following is equal to the expression listed below?
18 + 12
OA. 6(3+2)
OB. (6 x 3)(6 x 2)
OC. 6+ (3 x 2)
OD. (6 + 3)(6 + 2)
Step-by-step explanation:
the answer is (B)
True or False
_____ The ANOVA Test uses the entire bell
_____ There are 2 types of hypotheses
_____ The null hypothesis may posit that there is no significant difference betwe
The given question can be answered as follows:
True or False:
The ANOVA Test uses the entire bell. - False_
____ There are 2 types of hypotheses. - True_
____ The null hypothesis may posit that there is no significant difference between - True
Explanation:
ANOVA (Analysis of Variance) is a statistical test used to determine whether two or more population means are equivalent. It does not use the entire bell, so the statement is false.There are two types of hypotheses: the null hypothesis and the alternative hypothesis. The statement is true.The null hypothesis is used to determine if there is a significant difference between two or more sets of data. It can posit that there is no significant difference between two or more sets of data. So, the statement is true.
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The ANOVA Test uses the entire bell is a false statement.
There are 2 types of hypotheses is a true statement.
The null hypothesis may posit that there is no significant difference between the groups is a true statement.
Hence, the correct answer is: False; True; True.
The given statements can be summarized as follows: Statement 1 is that the ANOVA Test uses the entire bell. It is false.
Statement 2 is that there are 2 types of hypotheses. It is True.
Statement 3 is that the null hypothesis may posit that there is no significant difference between the groups. It is True.
Hence, the correct answer is: False; True; True.
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6. Select the coldest temperature from the list below. * O -11°F O 7°F O -4°F O 20°F
Answer:
-11 degrees Fahrenheit.
Step-by-step explanation:
the higher the negative number is, the colder it gets <D
i F 49x ²b = (7x +1/2)(7x-1/2)
then Find the value b.
Answer:
0
Step-by-step explanation:
Apply Only the Outside and Inside Method of the Foil Method.
[tex]7 \times - \frac{1}{2} = - 3.5x[/tex]
[tex] \frac{1}{2} \times 7x = 3.5x[/tex]
Add them together
[tex] - 3.5x + 3.5x = 0[/tex]
So our b value is 0.
Given function f(x) = (x + x)(log(x) + 3x) a) Show O(x) for f(x). You must show the obtained witnesses and k such that f) C(x) whenever x > k.(5 points) b) Show (x) for f(x). You must show the obtained witnesses and k such that f(x) Clg(x) whenever x > k. (5 points) c) What are the obtained witnesses C. C, and k such that Glg(x) (x) < Calg(x) whenever * >k. What is g(x).
a. For the function f(x) = (x + x)(log(x) + 3x) is O(x) with the witnesses C = 7 and k = 1.
b. It is proved that f(x) = (x + x)(log(x) + 3x) is (x) with the witnesses C = 1 and k = 1.
c. The obtained witnesses C = 1, C' = 10, and k =[tex]10^C.[/tex] such that g(x)log(x) < Cg(x) whenever x > k. And g(x) = 1.
a) To show that f(x) = (x + x)(log(x) + 3x) is O(x),
find witnesses C and k such that f(x) ≤ C × x for all x > k.
Let's simplify the expression for f(x):
f(x) = 2x × (log(x) + 3x)
= 2x × log(x) + 6x²
Now, find a witness C and a value k such that f(x) ≤ C × x for all x > k.
Let's choose C = 7 and k = 1.
This means show that f(x) ≤ 7x for all x > 1.
For x > 1,
f(x) = 2x × log(x) + 6x²
< 2x × log(x) + 6x² + 7x
= 2x × log(x) + 6x² + 7x
= x(2log(x) + 6x + 7)
≤ x(2log(x) + 13x)
≤ x × 7
= 7x
This implies,
f(x) = (x + x)(log(x) + 3x) is O(x) with the witnesses C = 7 and k = 1.
b) To show that f(x) = (x + x)(log(x) + 3x) is (x),
find witnesses C and k such that f(x) ≥ C × x for all x > k.
Let us simplify the expression for f(x):
f(x) = 2x × (log(x) + 3x)
= 2x × log(x) + 6x²
Now, find a witness C and a value k such that f(x) ≥ C × x for all x > k.
Let us choose C = 1 and k = 1.
This means show that f(x) ≥ x for all x > 1.
For x > 1,
f(x) = 2x × log(x) + 6x²
> x × log(x) + 6x²
= x(log(x) + 6x)
≥ x(log(x) + x)
≥ x × log(x)
≥ x
Therefore, we have shown that f(x) = (x + x)(log(x) + 3x) is (x) with the witnesses C = 1 and k = 1.
c) To find the obtained witnesses C, C', and k .
such that g(x)log(x) < Cg(x) whenever x > k,
Examine the expression f(x) = (x + x)(log(x) + 3x) and determine the function g(x).
Let us simplify the expression for f(x),
f(x) = 2x × (log(x) + 3x)
= 2x × log(x) + 6x²
From the given condition, we have g(x)log(x) < Cg(x) rewrite this as,
log(x) < C
Since log(x) is an increasing function, if log(x) < C, it means x < [tex]10^C.[/tex]Therefore, the witness k is [tex]10^C.[/tex]
Now let us determine g(x).
Since g(x)log(x) appears in the inequality, we can take g(x) = 1.
C = 1, C' = 10, and k = [tex]10^C.[/tex] for g(x)log(x) < Cg(x) whenever x > k. And g(x) = 1.
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A zoo has 5 Emperor penguins. The Emperor penguins make up 30%, percent of all the penguins at the zoo. How many penguins live at the zoo
Answer:
50
Step-by-step explanation:
this can be solved by ratio 15/x = 30/100. Cross multiply and solve for x. The answer is 50.
Joe's lunch at a restaurant cost $18.00 without tax he leaves the server a tip of 14% of the cost of lunch without tax what is the total cost of lunch including tip without tax
Ok. So 14% of 18.00 can be figured out without a calculator. 18.00 / 10 = 1.80 = 10%.
(18.00 / 100) x 4 = 0.72
1.80 + 0.72 = $2.52 is the tip.
$18 + $2.52 = $20.52
⭐ Answered by Foxzy0⭐
⭐ Brainliest would be appreciated, I'm trying to reach genius! ⭐
⭐ If you have questions, leave a comment, I'm happy to help! ⭐
Two level factorial experimentation led to the following model of the manufacturing yield (in terms of percentage of good products): ỹ = 95 +3.5x, -1.6x, +0.9x,x, where x, denotes reactor temperature, while X2 denotes air pressure. Experimental noise was estimated to be o, = 0.6. a) What are the main and interaction effects of the relevant variables. (6 points) b) During experimentation, the temperature was varied between 300K and 320K, while pressure was varied between 100kPa and 200kPa. Estimate the probability that the yield would be higher than 96% for temperature of 315K and pressure of 130kPa?
The main effects of the relevant variables are 3.5x and -1.6x, while the interaction effect is 0.9x*x.
The given model for manufacturing yield, expressed as a percentage of good products, is represented by the equation ỹ = 95 + 3.5x - 1.6x + 0.9x*x. In this equation, x represents the reactor temperature, and X2 represents the air pressure.
The coefficient 3.5 corresponds to the main effect of the reactor temperature, indicating that for each unit increase in temperature, the yield is expected to increase by 3.5 percentage points.
Similarly, the coefficient -1.6 represents the main effect of air pressure, implying that for each unit increase in pressure, the yield is expected to decrease by 1.6 percentage points.
Additionally, the term 0.9x*x accounts for the interaction effect between temperature and pressure. This suggests that the combined influence of temperature and pressure on the yield is not solely determined by the sum of their individual effects. Instead, the interaction effect captures the nonlinear relationship between these variables.
To estimate the probability of the yield being higher than 96% for a temperature of 315K and pressure of 130kPa, we need to evaluate the model equation for these specific values.
Substituting x = 315 and X2 = 130 into the equation, we can calculate the corresponding yield. If the yield exceeds 96%, the estimated probability would be 100%; otherwise, it would be 0%.
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Find the value of x that makes the equation true:
2x = 24
x = 6
x = 4
x = 9
x = 12
Answer:
x = 12 the correct answer
What value of x makes this proportion true? x/6=36/24 O A. 8 O B. 9 O C. 18 O D. 7
Answer:
B. 9
Step-by-step explanation:
Proportions are just like fractions and to figure them out sometimes you can use simplification in different forms.
[tex]\frac{x}{6}[/tex] = [tex]\frac{36}{24}[/tex]
Now to get to 6, 24 had to be divided by 4...in proportions and fractions usually, the top and bottom are both simplified or proportioned to the same number or scale
36 ÷ 4 = 9
Check your answer by inserting it there to see if it works
9*4 = 36 6*4 = 24
Answer:
[tex]\boxed {\boxed {\sf B. \ 9}}[/tex]
Step-by-step explanation:
We are given this proportion:
[tex]\frac {x}{6}=\frac{36}{24}[/tex]
We want to solve for x, so we must isolate the variable using inverse operations.
It is being divided by 6. The inverse of division is multiplication, so we multiply both sides of the proportion by 6.
[tex]6*\frac {x}{6}=\frac{36}{24}*6[/tex]
[tex]x=\frac{36}{24}*6[/tex]
[tex]x=1.5*6 \\x=9[/tex]
Another way to solve is with cross multiplication. Multiply the first numerator by the second denominator, then the first denominator by the second numerator.
[tex]\frac { x}{6}=\frac{36}{24}[/tex]
[tex]24*x=6*36[/tex]
[tex]24x=216[/tex]
The variable is being multiplied by 24. The inverse of multiplication is division, so we divide both sides of the equation by 24.
[tex]24x/24=216/24\\x=9[/tex]
The value of x that makes this proportion true is 9.
Let P be the vector space of polynomials of degree at most 2. Select each subset of P that is a subspace. Explain your reasons. (No credit for an answer alone.) (a) {p(x) = P₂|p(2)=0} (b) {p(z) € P₂ | x-p'(x) + p(x) = 0} (c) {p(z) E P₂|p(0) = P(1)} d) {ar2 + (a +1)x+b|a, b ER}
Let P be the vector space of polynomials of degree at most.
(a) is a subspace of P.
(b) is not a subspace of P.
(c) is a subspace of P.
(d) is a subspace of P.
(a) {p(x) = P₂|p(2)=0}
This subset consists of polynomials in P₂ that evaluate to 0 at x = 2. To check if it is a subspace, we need to verify the three conditions:
The zero polynomial is in this subset since it evaluates to 0 at x = 2.
Let p₁(x) and p₂(x) be two polynomials in this subset. If p₁(2) = 0 and p₂(2) = 0, then (p₁ + p₂)(2) = p₁(2) + p₂(2) = 0 + 0 = 0. Hence, the subset is closed under vector addition.
Let p(x) be a polynomial in this subset, and c be a scalar. If p(2) = 0, then (cp)(2) = c(p(2)) = c(0) = 0. Hence, the subset is closed under scalar multiplication.
Therefore, (a) is a subspace of P.
(b) {p(z) € P₂ | x-p'(x) + p(x) = 0}
This subset consists of polynomials in P₂ that satisfy the equation x - p'(x) + p(x) = 0. To check if it is a subspace, we need to verify the three conditions:
The zero polynomial is not in this subset since it does not satisfy the equation x - p'(x) + p(x) = 0.
If p₁(x) and p₂(x) are two polynomials in this subset, (p₁ + p₂)(x) = p₁(x) + p₂(x) satisfies the equation x - (p₁ + p₂)'(x) + (p₁ + p₂)(x) = 0. However, we need to check if it satisfies the equation x - (p₁ + p₂)'(x) + (p₁ + p₂)(x) = 0 for all x, not just at certain points. This condition may not hold, so the subset is not closed under vector addition.
Let p(x) be a polynomial in this subset, and c be a scalar. If we consider cp(x), the equation x - (cp)'(x) + cp(x) = 0 may not hold for all x, depending on the value of c. Therefore, the subset is not closed under scalar multiplication.
Therefore, (b) is not a subspace of P.
(c) {p(z) E P₂|p(0) = p(1)}
This subset consists of polynomials in P₂ that satisfy the equation p(0) = p(1). To check if it is a subspace, we need to verify the three conditions:
The zero polynomial is in this subset since it satisfies the equation p(0) = p(1) (both sides are 0).
If p₁(x) and p₂(x) are two polynomials in this subset, (p₁ + p₂)(x) = p₁(x) + p₂(x) satisfies the equation (p₁ + p₂)(0) = p₁(0) + p₂(0) and (p₁ + p₂)(1) = p₁(1) + p₂(1). Since p₁(0) = p₁(1) and p₂(0) = p₂(1), it follows that (p₁ + p₂)(0) = (p₁ + p₂)(1). Hence, the subset is closed under vector addition.
Let p(x) be a polynomial in this subset, and c be a scalar. If we consider cp(x), the equation (cp)(0) = (cp)(1) holds since p(0) = p(1). Hence, the subset is closed under scalar multiplication.
Therefore, (c) is a subspace of P.
(d) {ar² + (a + 1)x + b | a, b ∈ R}
This subset consists of all polynomials of the form ar² + (a + 1)x + b, where a and b are real numbers. To check if it is a subspace, we need to verify the three conditions:
The zero polynomial is in this subset since it can be written as 0r² + (0 + 1)x + 0 = x.
If p₁(x) and p₂(x) are two polynomials in this subset, their sum p₁(x) + p₂(x) is of the form ar² + (a + 1)x + b, where a and b are real numbers. Hence, the subset is closed under vector addition.
Let p(x) be a polynomial in this subset, and c be a scalar. Then cp(x) is of the form car² + (ca + c)x + cb, where a, b, and c are real numbers. Hence, the subset is closed under scalar multiplication.
Therefore, (d) is a subspace of P.
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An endocrinologist is interested in the effects of depression on the thyroid. It is believed that healthy subjects have a mean thyroxin (a hormone related to thyroid function) level of 7.0 micrograms/100 ml and a standard deviation of 1.6 micrograms/100 ml. The endocrinologist wants to assess whether the mean thyroxin level is different for those with depression. She samples 35 subjects with depression and obtains a sample mean of 7.82 micrograms/100 ml for thyroxin. What null and alternative hypotheses should she test
Answer:
H0: μ = 7.0
H1: μ > 7.0
Step-by-step explanation:
The null hypothesis will equal and take up the value of the population mean value ;
The population mean value, μ is 7.0
Null hypothesis ; H0: μ = 7.0
The alternative hypothesis will align with the claim ; which will take up and side with the value and direction of the sample mean ; 7.82 micrograms/100 ml
7.82 > 7.0 (sample mean is greater Than the population mean).
Hence, the alternative hypothesis, H1 will be ;
H1 : μ > 7.0